The values of wave direction and wavelength that are presented for 

 refraction by shear currents are correct. The paper contains 

 photographs of the entrance to Humboldt Bay which gives an indication of 

 the increase in wavelength with a following current, and the increase in 

 steepness on an adverse current under natural conditions. 



Coastal Engineering Significance. This is one of three papers included 

 in the annotated bibliography for its historical significance. Although 

 superseded and corrected by later work, it has a useful simplicity and 

 directness. 



23. JONSSON, I.G., "The Friction Factor for a Current Superimposed by 

 Waves," Progress Report No. 11, Coastal Engineering Laboratory and 

 Hydraulic Laboratory, Technical University of Denmark, Copenhagen, 

 Denmark, Apr. 1966, pp. 2-12. 



Keywords. Bottom Friction; Comparison of Theory and Measurement; 

 Conservation Equations; Currents, Unidirectional; Radiation Stress; 

 Setdown; Setup; Theory; Turbulence; Wave Dissipation; Wave Effect on 

 Current . 



Discussion. Plane turbulent flow over a horizontal, rough bottom is 

 considered. The current velocity is assumed uniformly distributed over 

 depth and not to exceed the bottom particle velocity in the wave motion. 

 Further the Froude number is assumed small. Both waves and current are 

 steady. 



For a pure wave motion over a horizontal bed, the reduction in 

 radiation stress in the direction of wave travel will cause a small 

 setup of the mean water surface. From the momentum equation for the 

 combined current wave motion it is demonstrated, however, that even a 

 very small current velocity — order of magnitude Froude number 0.01 — 

 will produce a negative tilt of the mean water surface, a setdown. This 

 setdown occurs once the downward slope of the surface needed to drive 

 the current exceeds the setup due to the waves. 



The instantaneous bed shear stress is assumed proportional with the 

 instantaneous total particle velocity (wave plus current) squared, and 

 phase differences are neglected. 



The result of the momentum equation is that measurement of wave 

 height gradient and mean water surface slope will give the current 

 friction factor. The energy equation, however, makes it possible to 

 eliminate either of the two quantities. 



In the formulation of the energy equation, it is pointed out that 

 the reference level for the potential energy must be horizontal. This 

 adds an extra term to the conventional expression for the energy flux 

 (at the time of the paper, wave action had not yet been introduced in 

 water wave dynamics). A strict physical interpretation of the new 



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